PDF. Introduction to Probability Distributions

2007 ◽  
pp. 55-68 ◽  
Author(s):  
Stacey J. Shaefer ◽  
Louis Theodore
Keyword(s):  
Probability Distributions

Asteroid and Comet Impact Speeds upon Mars: Significance for Panspermia and the Supply of Organics

1997 ◽  
Vol 161 ◽  
pp. 197-201 ◽  
Author(s):  
Duncan Steel
Keyword(s):  
Probability Distributions ◽  
Regions Of Interest ◽  
Significant Fraction ◽  
Organic Chemicals ◽  
Impact Speed ◽  
The Mean ◽  
The Impact

AbstractWhilst lithopanspermia depends upon massive impacts occurring at a speed above some limit, the intact delivery of organic chemicals or other volatiles to a planet requires the impact speed to be below some other limit such that a significant fraction of that material escapes destruction. Thus the two opposite ends of the impact speed distributions are the regions of interest in the bioastronomical context, whereas much modelling work on impacts delivers, or makes use of, only the mean speed. Here the probability distributions of impact speeds upon Mars are calculated for (i) the orbital distribution of known asteroids; and (ii) the expected distribution of near-parabolic cometary orbits. It is found that cometary impacts are far more likely to eject rocks from Mars (over 99 percent of the cometary impacts are at speeds above 20 km/sec, but at most 5 percent of the asteroidal impacts); paradoxically, the objects impacting at speeds low enough to make organic/volatile survival possible (the asteroids) are those which are depleted in such species.

An Accelerated Cue Combination Principle Accounts for Multi-cue Depth Perception

2020 ◽  
Vol 3 (1) ◽  
pp. 10501-1-10501-9
Author(s):  
Christopher W. Tyler
Keyword(s):  
Belief Propagation ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Fusion Model ◽  
Cue Combination ◽  
Depth Estimate ◽  
Appropriate Behavior ◽  
Depth Cues ◽  
Bayesian Averaging

Abstract For the visual world in which we operate, the core issue is to conceptualize how its three-dimensional structure is encoded through the neural computation of multiple depth cues and their integration to a unitary depth structure. One approach to this issue is the full Bayesian model of scene understanding, but this is shown to require selection from the implausibly large number of possible scenes. An alternative approach is to propagate the implied depth structure solution for the scene through the “belief propagation” algorithm on general probability distributions. However, a more efficient model of local slant propagation is developed as an alternative.The overall depth percept must be derived from the combination of all available depth cues, but a simple linear summation rule across, say, a dozen different depth cues, would massively overestimate the perceived depth in the scene in cases where each cue alone provides a close-to-veridical depth estimate. On the other hand, a Bayesian averaging or “modified weak fusion” model for depth cue combination does not provide for the observed enhancement of perceived depth from weak depth cues. Thus, the current models do not account for the empirical properties of perceived depth from multiple depth cues.The present analysis shows that these problems can be addressed by an asymptotic, or hyperbolic Minkowski, approach to cue combination. With appropriate parameters, this first-order rule gives strong summation for a few depth cues, but the effect of an increasing number of cues beyond that remains too weak to account for the available degree of perceived depth magnitude. Finally, an accelerated asymptotic rule is proposed to match the empirical strength of perceived depth as measured, with appropriate behavior for any number of depth cues.

Estimation of Regional Parameters in a Macro Scale Hydrological Model

2001 ◽  
Vol 32 (3) ◽  
pp. 161-180 ◽  
Author(s):  
Kolbjørn Engeland ◽  
Lars Gottschalk ◽  
Lena Tallaksen
Keyword(s):  
Additional Data ◽  
Hydrological Model ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Repeated Application ◽  
Landscape Characteristics ◽  
Time Period ◽  
Macro Scale ◽  
Model Element ◽  
Regular Sampling

Macro-scale hydrological modelling implies a repeated application of a model within an area using regional parameters. These parameters are based on climate and landscape characteristics, and they are used to calculate the water balance in ungauged areas. The regional parameters ought to be robust and not too dependent of the catchment and time period used for calibration. The ECOMAG model is applied for the NOPEX-region as a macro-scale hydrological model distributed on a 2×2 km2 grid. Each model element is assigned parameters according to soil and vegetation classes. A Bayesian methodology is followed. An objective function describing the fit between observed and simulated values is used to describe the likelihood of the parameters. Using Baye's theorem these likelihoods are used to update the probability distributions of the parameters using additional data, being it either an additional year of streamflow or an additional streamflow station. Two sampling methods are used, regular sampling and Metropolis-Hastings sampling. The results show that regional parameters exist according to some predefined criteria. The probability distribution of the parameters shows a decreasing variance as data from new catchments are used for updating. A few parameters do, however, not exhibit this property, and they are therefore not suitable in a regional context.

Investigation of a New Approach to Predict Water Quality Extremes with a Case Study of Chemical Determinands in Stream Water

1991 ◽  
Vol 24 (6) ◽  
pp. 25-33
Author(s):  
A. J. Jakeman ◽  
P. G. Whitehead ◽  
A. Robson ◽  
J. A. Taylor ◽  
J. Bai
Keyword(s):  
Water Quality ◽  
Stream Water ◽  
Probability Distributions ◽  
Management Strategies ◽  
Hybrid Modelling ◽  
Trend Model ◽  
Environmental Extremes ◽  
Modelling Approach

The paper illustrates analysis of the assumptions of the statistical component of a hybrid modelling approach for predicting environmental extremes. This shows how to assess the applicability of the approach to water quality problems. The analysis involves data on stream acidity from the Birkenes catchment in Norway. The modelling approach is hybrid in that it uses: (1) a deterministic or process-based description to simulate (non-stationary) long term trend values of environmental variables, and (2) probability distributions which are superimposed on the trend values to characterise the frequency of shorter term concentrations. This permits assessment of management strategies and of sensitivity to climate variables by adjusting the values of major forcing variables in the trend model. Knowledge of the variability about the trend is provided by: (a) identification of an appropriate parametric form of the probability density function (pdf) of the environmental attribute (e.g. stream acidity variables) whose extremes are of interest, and (b) estimation of pdf parameters using the output of the trend model.

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